$W(t)$ models the daily water level (in $\text{cm}$ ) at a pond in Arizona, $t$ days after the hottest day of the year. Here, $t$ is entered in radians. $W(t) = 15\cos\left(\dfrac{2\pi}{365}t\right) + 43$ What is the first time after the hottest day of the year that the water level is $30 \text{ cm}$ ? Round your final answer to the nearest whole day.
Answer: Converting the problem into mathematical terms $W(t) = 15\cos\left({\dfrac{2\pi}{365}}t\right) + 43$ has a period of $\dfrac{2\pi}{{\scriptsize\dfrac{2\pi}{365}}}=365$ days. We want to find the first solution to the equation $W(t)=30$ within the period $0<t<365$. The answer The equation's two solutions within the desired period (rounded to the nearest whole day) are $152$ and $213$. Therefore, the first time that the water level hits $30 \text{ cm}$ is after $152$ days after the hottest day of the year.